For a Sato-Tate group $G$ of degree $d$ and weight $w$ a **Hodge circle** is a continuous homomorphism
\[
h\colon \mathrm{U}(1)\to G^0
\]
for which there exist nonnegative integers $h^{p,q}$ satisfying $\sum h^{p,q}=d$ with $p,q$ varying over integers in $[0,w]$ such that

- $h(u)$ has eigenvalues $u^{p-q}$ with multiplicity $h^{p,q}$ for all $u\in \mathrm{U}(1)$;
- the conjugates of the image of $h$ generate a dense subgroup of $G^0$.

When $G$ is the Sato-Tate group of a motive $X$ of weight $w$, the $h^{p,q}$ are the Hodge numbers of $X$. For a $d$-dimensional abelian variety $X$ we necessarily have $h^{0,1}=h^{1,0}=d$.

**Authors:**

**Knowl status:**

- Review status: beta
- Last edited by Andrew Sutherland on 2021-01-16 14:47:48

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**History:**(expand/hide all)

- 2021-01-16 14:47:48 by Andrew Sutherland
- 2021-01-16 14:47:21 by Andrew Sutherland
- 2021-01-15 12:22:42 by Andrew Sutherland

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